Finite Rank Perturbations of Locally Definitizable Self-adjoint Operators in Krein Spaces. Jussi Behrndt

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1 Finite Rank Perturbations of Locally Definitizable Self-adjoint Operators in Krein Spaces Jussi Behrndt Abstract. In this paper we generalize the well-known result that a definitizable operator in a Krein space remains definitizable after a finite dimensional perturbation to a class of self-adjoint operators in Krein spaces which locally have the same spectral properties as definitizable operators. Keywords: self-adjoint operators in Krein spaces, finite rank perturbations, locally definitizable operators, spectral points of positive and negative type, self-adjoint extensions of symmetric operators MSC 2000: 47B50, 47A55, 47B40 1. Introduction A self-adjoint operator A in a Krein space K, [, ] is called definitizable if the resolvent set ρa is nonempty and there exists a polynomial p such that [pax, x] 0 for all x dom pa. Spectral and perturbation theory of definitizable operators is well developed and of great importance in many applications see e.g. [6, 7, 8, 14, 25, 29, 30, 31, 34]. It was shown by H. Langer in [30, 31] see also [18] that a definitizable operator A has a spectral function and with the help of this spectral function the real points of the spectrum σa can be classified in points of positive and negative type and a finite set of so-called critical points. A fundamental paper on perturbations of definitizable operators is [25] by P. Jonas and H. Langer where it is proved that a definitizable operator remains definitizable after a finite dimensional perturbation in resolvent sense if the perturbed operator is self-adjoint and has a nonempty resolvent set. The aim of this paper is to generalize this perturbation result to a class of self-adjoint operators in Krein spaces which locally have the same spectral properties as definitizable operators. More precisely, let Ω be some domain in C symmetric with respect to the real line such that Ω R and the intersections

2 of Ω with the upper and lower open half-planes are simply connected. A selfadjoint operator A is said to be definitizable over Ω if every point µ Ω R has an open connected neighbourhood I µ in R such that the spectral points in each component of I µ \{µ} are all of the same sign type, the nonreal spectrum of A in Ω\R does not accumulate to Ω R, consists of isolated points which are poles of the resolvent of A and the resolvent is of finite order growth near to Ω R cf. [23]. Perturbations of locally definitizable operators and stability properties of spectral points of positive and negative type and socalled spectral points or intervals of type π + and type π were investigated in e.g. [2, 4, 19, 20, 21, 33]. The main result of this note is Theorem 3.2 where we prove that a selfadjoint operator, or more generally a self-adjoint relation, which is locally definitizable over Ω remains locally definitizable over Ω after a finite dimensional perturbation in resolvent sense if the perturbed operator or relation is self-adjoint and the unperturbed and perturbed operator or relation have a common point in their resolvent sets belonging to Ω. For the special case of definitizable operators this result coincides with [25, Theorem 1] mentioned above. The methods used in the proof of Theorem 3.2 differ from those applied in [25]. Our proof is based on a variant of [4, Theorem 2.4] see Theorem 3.1 on the stability of intervals of type π + and type π under compact perturbations and a recent result from [1] on the spectral properties of the inverses of certain matrix-valued functions associated to locally definitizable operators and relations. We briefly describe the contents of this paper. In Section 2 we introduce the spectral points of positive and negative type with the help of approximative eigensequences and we recall the definitions and connections between locally definitizable self-adjoint relations and locally definitizable functions from [23] and [24]. In particular, Theorem 2.8 on the representation of a locally definitizable function with the help of the resolvent of a locally definitizable self-adjoint relation is an important tool in the proof of our main result Theorem 3.2 which is the focus of Section 3. In Section 4.1 we apply our perturbation result to the self-adjoint extensions of symmetric operators or relations of finite defect. We use the concept of boundary value spaces and associated Weyl functions for the parametrization of the closed extensions of a symmetric relation and the description of their spectral properties see e.g. [10, 11, 12, 13]. As an example we consider in Section 4.2 the direct sum of a regular and a singular Sturm-Liouville differential operator with the indefinite weight sgn x and we show in Section 4.3 that in such a general setting self-adjoint differential operators with an empty resolvent set can appear. 2

3 2. Locally definitizable self-adjoint relations and locally definitizable functions 2.1. Preliminaries. The linear space of bounded linear operators defined on a Krein space K 1 with values in a Krein space K 2 is denoted by LK 1, K 2. If K := K 1 = K 2 we simply write LK. We study linear relations from K 1 to K 2, that is, linear subspaces of K 1 K 2. The set of all closed linear relations from K 1 to K 2 is denoted by CK 1, K 2. If K = K 1 = K 2 we write CK. Linear operators from K 1 into K 2 are viewed as linear relations via their graphs. For the usual definitions of the linear operations with relations, the inverse etc., we refer to [15]. The sum and the direct sum of subspaces in K 1 K 2 will be denoted by and.. In the following let K, [, ] be a separable Krein space and let S be a closed linear relation in K. The resolvent set ρs of S is the set of all λ C such that S λ 1 LK, the spectrum σs of S is the complement of ρs in C. We say that λ C is a point of regular type of S, λ rs, if S λ 1 is a bounded operator. A point λ C is an eigenvalue of S if kers λ {0}; we write λ σ p S. We say that λ C belongs to the continuous spectrum σ c S the residual spectrum σ r S of S if kers λ = {0} and ran S λ is dense in K resp. if kers λ = {0} and ran S λ is not dense in K. We say that λ C belongs to the approximate point spectrum of S, denoted by σ ap S, if there exists a sequence x n yn S, n = 1, 2,..., such that xn = 1 and lim n y n λx n = 0. The extended approximate point spectrum σ ap S of S is defined by σ ap S := { σ ap S { } if 0 σ ap S 1 σ ap S if 0 σ ap S 1. Next we recall the definitions of the spectra of positive and negative type of a closed linear relation see e.g. [23, 33]. Definition 2.1. Let S be a closed linear relation in K. A point λ σ ap S is said to be of positive type negative type with respect to S, if for every sequence x n y n S, n = 1, 2..., with xn = 1, lim n y n λx n = 0 we have lim inf n [x n, x n ] > 0 resp. lim sup n [x n, x n ] < 0. If σ ap S, is said to be of positive type negative type with respect to S if for every sequence x n y n S, n = 1, 2..., with limn x n = 0, y n = 1 3

4 we have lim inf n [y n, y n ] > 0 resp. lim sup n [y n, y n ] < 0. The set of all points of positive type negative type with respect to S will be denoted by σ ++ S resp. σ S. An open subset of R is said to be of positive type negative type with respect to S if σs σ ++ S resp. σs σ S. An open subset of R is called of definite type with respect to S if is of positive or negative type with respect to S. Note, that σ ap S is of positive negative type with respect to S if and only if 0 is of positive resp. negative type with respect to S 1. Let S be a linear relation in K. The adjoint relation S + CK is defined by { } h f S + := [h, f h ] = [h, f] for all f S. S is said to be symmetric self-adjoint if S S + resp. S = S +. We remark that for a self-adjoint relation S in K the points of definite type introduced in Definition 2.1 belong to R Locally definitizable self-adjoint relations. Let Ω be some domain in C symmetric with respect to the real axis such that Ω R and the intersections of Ω with the upper and lower open half-planes are simply connected. Let A be a self-adjoint relation in the Krein space K such that σa Ω\R consists of isolated points which are poles of the resolvent of A, and no point of Ω R is an accumulation point of the nonreal spectrum of A in Ω. Let be an open subset of Ω R. We say that A belongs to the class S, if for every finite union of open connected subsets,, there exists m 1, M > 0 and an open neighbourhood U of in C such that 2.1 A λ 1 M1 + λ 2m 2 Im λ m holds for all λ U\R. The next definition can be found in e.g. [20]. Definition 2.2. Let Ω be a domain as above and let A be a self-adjoint relation in the Krein space K such that σa Ω\R consists of isolated points which are poles of the resolvent of A and no point of Ω R is an accumulation point of the nonreal spectrum of A in Ω. The relation A is said to be definitizable over Ω, if A S Ω R and every point µ Ω R has an open connected neighbourhood I µ in R such that both components of I µ \{µ} are of definite type with respect to A. 4

5 By [23, Theorem 4.7] a self-adjoint relation A is definitizable over C if and only if A is definitizable, that is, the resolvent set of A is nonempty and there exists a real polynomial p such that [ pax, x ] 0 holds for all x dom pa. For a detailed study of definitizable self-adjoint operators and relations we refer to the fundamental paper [31] of H. Langer and to [16, 4 and 5]. The next theorem is a simple modification of [23, Theorem 4.8]. Theorem 2.3. Let A be a self-adjoint relation in K and let Ω be a domain as above. A is definitizable over Ω if and only if for every domain Ω with the same properties as Ω, Ω Ω, there exists a self-adjoint projection E in K such that A can be decomposed in A = A EK 2. A 1 EK 2 and the following holds. i A EK 2 is a definitizable relation in the Krein space EK. ii σ A 1 EK 2 Ω =. Let A = A + be definitizable over Ω, let Ω be a domain with the same properties as Ω, Ω Ω, and let E be a self-adjoint projection with the properties as in Theorem 2.3. If E is the spectral function of the definitizable self-adjoint relation A EK 2 in the Krein space EK, then the mapping δ E δe =: E A δ defined for all finite unions δ of connected subsets of Ω R the endpoints of which belong to Ω R and are of definite type with respect to A EK 2, is the spectral function of A on Ω R see [23, Section 3.4 and Remark 4.9]. With the help of the local spectral function E A the open subsets of definite type in Ω R can be characterized in the following way. An open subset Ω R is of positive type negative type with respect to A if and only if for every finite union δ of open connected subsets of, δ, such that the boundary points of δ in R are of definite type with respect to A the spectral subspace E A δk, [, ] resp. E A δk, [, ] is a Hilbert space. As a generalization of open subsets of positive and negative type we introduce open subsets of type π + and type π in the next definition. Definition 2.4. Let Ω be a domain as in the beginning of this section and let A be a self-adjoint relation in K which is definitizable over Ω. An open subset of Ω R is said to be of type π + type π with respect to A if for every finite union δ of open connected subsets of, δ, such that the 5

6 boundary points of δ in R are of definite type with respect to A the spectral subspace E A δk, [, ] is a Pontryagin space with finite rank of negativity resp. positivity. We shall say that A is of type π + over Ω type π over Ω if Ω R is of type π + resp. type π with respect to A. We remark, that spectral points in open subsets of type π + and type π can also be characterized with the help of approximative eigensequences see [2] Matrix-valued locally definitizable functions. Let Ω be a domain as in the beginning of Section 2.2 and let τ be an LC n -valued piecewise meromorphic function in Ω\R which is symmetric with respect to the real axis, that is τλ = τλ for all points λ of holomorphy of τ. If, in addition, no point of Ω R is an accumulation point of nonreal poles of τ we write τ M n n Ω. The set of the points of holomorphy of τ in Ω\R and all points µ Ω R such that τ can be analytically continued to µ and the continuations from Ω C + and Ω C coincide, is denoted by hτ. The following definition of open sets of positive and negative type with respect to matrix functions and Definition 2.6 below of locally definitizable matrix functions can be found in [24]. Definition 2.5. Let τ M n n Ω. An open subset Ω R is said to be of positive type with respect to τ if for every x C n and every sequence µ k of points in Ω C + hτ which converges in C to a point of we have lim inf k Im τµ k x, x 0. An open subset Ω R is said to be of negative type with respect to τ if is of positive type with respect to τ. is said to be of definite type with respect to τ if is of positive or of negative type with respect to τ. Definition 2.6. A function τ M n n Ω is called definitizable in Ω if the following holds. i Every point µ Ω R has an open connected neighbourhood I µ in R such that both components of I µ \{µ} are of definite type with respect to τ. ii For every finite union of open connected subsets in R, Ω R, there exists m 1, M > 0 and an open neighbourhood U of in C such that holds for all λ U\R. τλ M1 + λ 2m Im λ m 6

7 If τ M n n C is definitizable in C there exists a scalar rational function g symmetric with respect to the real line such that the poles of g belong to hτ { } and gτ is the sum of a Nevanlinna function and a meromorphic function in C cf. [23, Theorem 4.7]. In this case we shall say that τ is a definitizable function. For a comprehensive study of definitizable functions we refer to [22]. A famous subclass of the definitizable functions are the generalized Nevanlinna functions introduced and studied by M.G. Krein and H. Langer see e.g. [28]. Recall that a function τ M n n C belongs to the class N κ, κ = 0, 1, 2,..., if the kernel K τ, K τ λ, µ := τλ τµ, λ µ has κ negative squares. Note, that the class N 0 coincides with the class of Nevanlinna functions. In [24] it is shown that a function τ M n n Ω is definitizable in Ω if and only if for every finite union of open connected subsets of R such that Ω R, τ can be written as the sum τ = τ 0 + τ 0 of an LC n -valued definitizable function τ 0 and an LC n -valued function τ 0 which is locally holomorphic on. Let τ M n n Ω. We shall say that an open subset Ω R is of type π + with respect to τ if for every open set δ which is the union of a finite number of pairwise disjoint connected open subsets of such that δ, τ can be written as the sum τ = τ 0 + τ 0 of an LC n -valued generalized Nevanlinna function τ 0 and an LC n -valued function τ 0 which is locally holomorphic on δ. We shall say that an open subset Ω R is of type π with respect to τ if is of type π + with respect to τ. The following theorem is a consequence of [24, 3.1] and [23, Theorem 3.18]. It establishes a connection between self-adjoint relations which are locally definitizable and LC n -valued locally definitizable functions. Theorem 2.7. Let Ω be a domain as above and let A be a self-adjoint relation in the Krein space K which is definitizable over Ω. Let γ LC n, K and S = S LC n, fix some point λ 0 ρa Ω and define τλ := S + γ + λ Re λ 0 + λ λ 0 λ λ 0 A λ 1 γ for all λ ρa Ω. Then the following holds. i The function τ is definitizable in Ω. ii An open subset of Ω R which is of positive type negative type with respect to A is of positive type resp. negative type with respect to τ. 7

8 iii An open subset of Ω R which is type π + type π with respect to A is of type π + resp. type π with respect to τ. Proof. In order to show assertions i and ii let µ Ω R and choose an open connected neighbourhood I µ of µ in R such that both components of I µ \{µ} are of definite type with respect to A. Assume e.g. that a component + of I µ \{µ} is of positive type with respect to A. Let x C n and let λ k be a sequence in Ω C + hτ which converges to some point in +. Making use of [23, Theorem 3.18] we obtain lim inf k Im τλ k x, x = lim inf k Im [ λ k Re λ 0 + λ k λ 0 λ k λ 0 A λ k 1 γ x, γ x ] 0 and this implies that + is of positive type with respect to τ. A similar reasoning shows that a component of I µ \{µ} which is of negative type with respect to A is also of negative type with respect to τ. Therefore property i of Definition 2.6 is fulfilled and assertion ii holds. The growth properties of the resolvent of A see 2.1 imply that τ is locally definitizable in Ω. It remains to prove assertion iii. Let δ be a finite union of open connected subsets of, Ω R, and choose a finite union δ 1 of open connected subsets of such that δ δ 1, δ 1 and the boundary points of δ 1 in R are of definite type with respect to A. As A is of type π + over the spectral subspace E A δ 1, [, ] is a Pontryagin space with finite rank of negativity. Therefore τ 0 λ := S + γ + λ Re λ 0 + λ λ 0 λ λ 0 A λ 1 E A δ 1 γ is a generalized Nevanlinna function and from σa 1 E A δ 1 K 2 δ = we obtain that τ 0 λ := γ + λ Re λ 0 + λ λ 0 λ λ 0 A λ 1 1 E A δ 1 γ is holomorphic in a neighbourhood of δ. Hence is of type π + with respect to τ. A similar argument shows that an open subset Ω R which is of type π with respect to A is also of type π with respect to τ. The next theorem states that a locally definitizable function can be represented with the help of the resolvent of a locally definitizable self-adjoint relation. A proof can be found in [24]. Theorem 2.8. Let τ be an LC n -valued function definitizable in Ω an LC n - valued local generalized Nevanlinna function in Ω and let Ω be a domain with the same properties as Ω such that Ω Ω. 8

9 Then there exists a Krein space G, a self-adjoint relation T in G which is definitizable over Ω resp. of type π + over Ω and a mapping γ LC n, G with the following properties. a ρt Ω = hτ Ω. b For a fixed λ 0 ρt Ω and all λ ρt Ω τλ = Re τλ 0 + γ + λ Re λ 0 + λ λ 0 λ λ 0 T λ 1 γ holds. c For any finite union of open connected subsets of R, Ω R, such that the boundary points of are of definite type with respect to τ the spectral projection E T is defined. If Ω is a domain with the same properties as Ω, Ω Ω, and E T Ω \R is the Riesz-Dunford projection corresponding to σt Ω \R and if we set E := E T + E T Ω \R, then the minimality condition EG = clsp { 1 + λ λ 0 T λ 1 Eγx λ ρt Ω, x C n} is fulfilled. d Any finite union of open connected subsets of R, Ω R, is of positive type negative type, type π +, type π with respect to τ if and only if is of positive type resp. negative type, type π +, type π with respect to T. If τ and T are as in Theorem 2.8 we shall say that T is an Ω -minimal representing relation for τ. 3. Finite rank perturbations of locally definitizable self-adjoint relations in Krein spaces In [25] P. Jonas and H. Langer proved that a self-adjoint definitizable operator remains definitizable after a finite dimensional perturbation in resolvent sense if the perturbed operator is self-adjoint and the unperturbed and perturbed operator have a common point in their resolvent sets. In this section we prove that this holds also for locally definitizable operators and relations. The methods we apply here differ from those used in the proof of [25, Theorem 1] where a definitizing polynomial for the perturbed operator was constructed. The essential ingredients in the proof of Theorem 3.2 below are [1, Theorem 2.5] which states that the inverse of a matrix-valued locally definitizable function is also locally definitizable, Theorem 2.8 on the representation of locally definitizable functions and a variant of [4, Theorem 2.4] on the stability of intervals of type π + under compact perturbations see Theorem

10 Let K, [, ] be a separable Krein space. The set of compact operators and finite rank operators defined on K with values in K will be denoted by S and F, respectively. Let, as in Section 2.2, Ω be some domain in C symmetric with respect to the real axis such that Ω R and the intersections of Ω with the upper and lower open half-planes are simply connected. The next theorem is a simple modification of [4, Theorem 2.4] see also [2, Theorem 29] and [33, Theorem 5.1]. Theorem 3.1. Let A and B be self-adjoint relations in the Krein space K, let ρa ρb Ω and assume that B λ 0 1 A λ 0 1 S holds for some λ 0 ρa ρb. Then A is definitizable over Ω and Ω R is of type π + type π with respect to A if and only if B is definitizable over Ω and Ω R is of type π + resp. type π with respect to B. The next theorem is the main result of this paper. Theorem 3.2. Let A and B be self-adjoint relations in the Krein space K, let ρa ρb Ω and assume that B λ 0 1 A λ 0 1 F holds for some λ 0 ρa ρb. Then A is definitizable over Ω if and only if B is definitizable over Ω. Moreover, if A is definitizable over Ω and Ω R is an open interval with endpoint µ Ω R and is of positive type negative type with respect to A, then there exists an open interval,, with endpoint µ such that is of positive type resp. negative type with respect to B. Proof. 1. Assume that A is a self-adjoint relation in K which is definitizable over Ω. Let λ 0 ρa ρb Ω and let e 1,..., e n, f 1,..., f n be vectors in K such that n 3.2 B λ 0 1 A λ 0 1 = [, e i ]f i. It is no restriction to assume that the system {f 1,..., f n } as well as the system {e 1,..., e n } is linearly independent. For λ ρa ρb the assumption λ 0 ρa ρb implies that the vectors 1 + λ λ0 A λ 1 f i, i = 1,..., n, 3.3 as well as the vectors 1 + λ λ0 B λ 1 e i, i = 1,..., n, i=1 10

11 and λ λ0 B λ 1 e i, i = 1,..., n, are also linearly independent. From 3.5 B λ 1 A λ 1 = 1 + λ λ 0 A λ 1 B λ0 1 A λ λ λ 0 B λ 1 we obtain that dimran B λ 1 A λ 1 < holds for every λ in ρa ρb. From the assumption that A is definitizable over Ω it follows that σa Ω\R consists of a discrete set of normal eigenvalues of A. Well known perturbation results imply 3.6 Ω\R ρb σ p,norm B. The set σ p,norm A σ p,norm B is discrete in Ω\R and we have Ω\R σp,norm A σ p,norm B ρa ρb. 3.7 Inserting 3.2 in 3.5 and using the self-adjointness of A and B yields B λ 1 A λ 1 n = [, 1 + λ λ0 B λ 1 ] e i 1 + λ λ0 A λ 1 f i = i=1 n [, 1 + λ λ0 A λ 1 ] f i 1 + λ λ0 B λ 1 e i i=1 for all λ ρa ρb. Replacing λ and λ 0 in 3.5 by λ and λ 0, respectively, and inserting the adjoint of 3.2 gives 3.8 B λ 1 A λ 1 n = [, 1 + λ λ0 A λ 1 ] e i 1 + λ λ0 B λ 1 f i i=1 for all λ ρa ρb. Let 3.9 By 3.8 we get 3.10 K := clsp { 1 + λ λ 0 A λ 1 f i K = clsp { 1 + λ λ 0 B λ 1 f i i = 1,..., n, λ ρa ρb Ω }. i = 1,..., n, λ ρa ρb Ω }. 11

12 If µ 0 ρa ρb Ω we have A µ 0 1 K K and A µ 0 1 K [ ] K [ ]. For x K [ ] 3.9 implies [ x, 1 + λ λ0 A λ 1 f i ] = 0 for all λ ρa ρb Ω and i = 1,..., n. Therefore 3.7 yields 3.11 A λ 1 K [ ] = B λ 1 K [ ], λ ρa ρb Ω. 2. In this part of the proof we show that there exists an invertible LC n - valued function α and mappings Γ λ0, Γ λ0 LC n, K such that αλ 1 = Re αλ Γ + λ 0 λ Re λ0 + and λ λ 0 λ λ 0 A λ 1 Γ λ0 αλ = Re αλ 0 + Γ + λ 0 λ Re λ0 + λ λ 0 λ λ 0 B λ 1 Γλ0 holds for all λ ρa ρb. Some of the following calculations can be found in [27, Proof of Theorem 5.1] and [35, Proof of Proposition 2.1]. For the convenience of the reader we present the details. By 3.3 and 3.4 the vectors 1 + λ λ 0 A λ 1 f j, j = 1,..., n, and 1 + λ λ 0 B λ 1 e i, i = 1,..., n, are linearly independent for every λ ρa ρb. Hence for every λ ρa ρb there exists an invertible matrix αλ = α ij λ n i,j=1 such that 1 + λ λ0 B λ 1 n e i = α ji λ 1 + λ λ 0 A λ 1 f j holds for all i = 1,..., n. Let Γ λ0 define j=1 : C n K, c 1,..., c n n i=1 c if i and Γ λ := 1 + λ λ 0 A λ 1 Γ λ0 for all λ ρa. Then Γ + : K λ Cn is given by [ ] x, 1 + λ λ0 A λ 1 f 1 and we can rewrite 3.7 as 3.12 x [. ] x, 1 + λ λ0 A λ 1 f n B λ 1 A λ 1 = Γ λ αλγ + λ. 12

13 Replacing λ by λ and taking adjoints in 3.12 we obtain Γ λ αλγ + λ = Γ λαλ Γ + λ. From ran Γ + λ = ker Γ λ and the fact that Γ λ and Γ λ are injective we conclude 3.13 αλ = αλ, It is straightforward to check that the relation λ ρa ρb. µ λ B λ 1 A λ 1 B µ 1 A µ 1 = 1 + λ µa λ 1 B µ 1 A µ 1 B λ 1 A λ µ λa µ 1 holds for all λ, µ ρa ρb cf. [27, Proof of Theorem 5.1]. Using 3.12 and Γ λ = 1 + λ µa λ 1 Γ µ, λ, µ ρa, we find µ λγ λ αλγ + λ Γ µαµγ + µ = Γ λαµγ + µ Γ λαλγ + µ. From ker Γ λ = ker Γ µ = {0} we obtain µ λαλγ + λ Γ µαµ = αµ αλ and 3.14 In particular the relation 3.15 µ λγ + λ Γ µ = αλ 1 αµ 1, λ, µ ρa ρb. Im λ 0 Γ + λ 0 Γ λ0 = Im αλ 0 1 holds. It is easy to see that the function τ defined for all λ ρa by 3.16 fulfils 3.17 λ τλ := Γ + λ 0 λ Re λ0 + λ λ 0 λ λ 0 A λ 1 Γ λ0 τλ τµ = λ µγ + λ Γ µ and τλ = τλ for all λ, µ ρa. The relations 3.13, 3.14, 3.15, 3.16 and 3.17 imply that the function λ αλ 1 + τλ is equal to the self-adjoint constant Reαλ 0 1 LC n. Therefore 3.12 can be written in the form B λ 1 = A λ 1 + Γ λ Re αλ 0 1 τλ 1Γ + λ 13

14 and 3.15 and 3.16 imply 3.18 αλ 1 = Re αλ Γ + λ 0 λ Re λ0 + λ λ 0 λ λ 0 A λ 1 Γ λ0 = αλ λ λ 0 Γ + λ λ λ0 A λ 1 Γ λ0 = αλ λ λ 0 Γ + λ λ0 λa λ Γλ0. Making use of 3.15 it follows that U := 1 λ 0 λ 0 Γ λ0 αλ 0 Γ + λ 0 LK is unitary, i.e. U + = U 1, and that UΓ λ0 αλ 0 = Γ λ0 αλ 0 holds. With the help of 3.12 and the relation λ 0 λ 0 Γ λ0 αλ 0 Γ + λ 0 Γ λ0 αλ 0 Γ + λ 0 = Γ λ0 αλ 0 Γ + λ 0 Γ λ0 αλ 0 Γ + λ 0, which follows easily from 3.15, we obtain that U1 + λ 0 λb λ 0 1 coincides with λ 0 λa λ 0 1 λ 0 λ 0 Γ λ0 αλ 0 Γ + λ 0 λ 0 λ 0 λ 0 λγ λ0 αλ 0 Γ + λ 0 A λ 0 1 +λ 0 λγ λ0 αλ 0 Γ + λ 0. From Γ + λ 0 = Γ + λ λ0 λ 0 A λ 0 1 and 3.19 we get U 1 + λ 0 λb λ 0 1 = 1 + λ 0 λa λ λ 0 λγ λ0 αλ 0 Γ + λ 0. In particular the right hand side is a boundedly invertible operator with an everywhere defined inverse. Now we can apply [32, Lemma 3.1] to 3.18 and we obtain αλ = αλ 0 + λ λ 0 αλ 0 Γ + λ λ0 λa λ λ 0 λγ λ0 αλ 0 Γ + λ 0 1 Γ λ0 αλ 0. Let Γ λ0 := Γ λ0 αλ 0. Then we have and therefore Γ λ0 = U 1 Γ λ0 αλ 0 and Im λ 0 Γ + λ 0 Γλ0 = Im αλ αλ = αλ 0 + λ λ 0 Γ + λ λ0 λb λ Γλ0 = αλ 0 + λ λ 0 Γ + λ λ λ0 B λ 1 Γλ0 = Re αλ 0 + Γ + λ 0 λ Re λ0 + λ λ 0 λ λ 0 B λ 1 Γλ0. 14

15 3. In this part of the proof we show that B is definitizable over Ω. We prove first that every point µ Ω R has an open connected neighbourhood I µ in R such that both components of I µ \{µ} are of definite type with respect to B. Let µ Ω R and assume e.g. that +, + Ω R, is an open interval with endpoint µ such that + is of positive type with respect to A. From Theorem 2.7 we obtain that the function αλ 1 = Re αλ Γ + λ 0 λ Re λ0 + λ λ 0 λ λ 0 A λ 1 Γ λ0 is definitizable in Ω and + is of positive type with respect to this function. By [1, Theorem 2.5] the function λ αλ is also definitizable in Ω and there exists a in general smaller smaller open interval +, + +, with endpoint µ, which is of positive type with respect to α. Let Ω + be a domain with the same properties as Ω such that Ω + Ω and + = Ω + R. As A is definitizable over Ω +, ρa ρb Ω + is nonempty and Ω + R is of positive type with respect to A we can apply Theorem 3.1. It follows that B is definitizable over Ω + and + is of type π + with respect to B. Let δ + be an open interval such that δ + + and E B δ + is defined. Then E A δ + K, [, ] is a Hilbert space and E B δ + K, [, ] is a Pontryagin space with finite rank of negativity. In the following we will show that E B δ + K equipped with the indefinite inner product [, ] is a Hilbert space. This will be done in four steps. i Let Ω be a domain with the same properties as Ω such that Ω Ω, + Ω R and λ 0 Ω holds. As the function α is definitizable in Ω and + is of positive type with respect to α we obtain from Theorem 2.8 that there exists a Krein space G, [, ] G, a self-adjoint relation T in G definitizable over Ω and a mapping Γ LC n, G such that ρt Ω = hα Ω and 3.21 αλ = Re αλ 0 + Γ + λ Re λ 0 + λ λ 0 λ λ 0 T λ 1 Γ holds for all λ ρt Ω. Note that by 3.20 the function α is holomorphic at λ 0 and therefore λ 0 belongs to ρt. According to Theorem 2.8 we can assume that T is chosen Ω -minimal and that + is of positive type with respect to T. Then the spectral projection E T δ + of T corresponding to the open interval δ + is defined, E T δ + G equipped with the inner product [, ] G is a Hilbert space and the condition E T δ + G = clsp { 1 + λ λ 0 T λ E T δ + Γx λ ρt Ω, x C n} is fulfilled. 15

16 From 3.20 and 3.21 we obtain Γ + λ 0 Γλ0 = Γ + Γ and Γ + λ 0 B λ 1 Γλ0 = Γ + T λ 1 Γ for all λ ρt ρb Ω. Therefore the relation V G K defined by { l V := k=1 1 + λ } k λ 0 T λ k 1 Γx k λ k ρt ρb Ω l k=1 1 + λ k λ 0 B λ k 1 Γ λ0 x k x k C n, k = 1,..., l is linear and isometric and the same holds for its closure V CG, K. ii Now we show that V is reduced by E T δ + G E B δ + K, i.e. we verify that V can be written as. I V E T δ + G E B δ + K V ET δ + G I E B δ + K. Let f g V and choose a sequence fm gm V such that fm gm fg for m. We assume first that the endpoints d 1 and d 2 of the bounded open interval δ + = d 1, d 2 are no eigenvalues of T and B. We fix some η > 0 such that the rectangle Q := { z C d 1 Re z d 2, η Im z η } has the property Q\R ρt ρb. Let the boundary C of Q be oriented in the mathematical positive sense and let the curves C k := C { z C Im z 1 k}, k > η 1, be oriented as C. As f m gm V, m = 1, 2,..., we obtain T λ 1 f m B λ 1 V g m for all λ ρt ρb Ω and m = 1, 2,.... Therefore the elements 1 2πi C k T λ 1 dλ f m 1 2πi C k B λ 1, m N, k > η dλ g 1, m belong to V. From and E T δ + f m = lim k 1 2πi C k T λ 1 dλ f m E B δ + g m = lim 1 B λ 1 dλ g m k 2πi C k 16

17 we conclude ET δ + f m ET δ + f E B δ + g E B δ + g m V and this implies V and I ET δ + f I E B δ + g V. Thus V is reduced by E T δ + G E B δ + K. If d 1 or d 2 is an eigenvalue of T or B it follows from the strong σ-additivity of the local spectral function and the case considered above that V is reduced by E T δ + G E B δ + K. iii We prove that V δ+ := V E T δ + G E B δ + K is an operator and that E B δ + K, [, ] cf. 3.9 and 3.10 is a Hilbert space. The relation V δ+ is isometric and by the definition of Γ λ0 and Γ λ0 we have ran Γ λ0 = ran Γ λ0 = sp {f i i = 1,..., n}. As the the elements l k=1 1 + λ k λ 0 T λ k 1 E T δ + Γx k l k=1 1 + λ, k λ 0 B λ k 1 E B δ + Γ λ0 x k λ k ρt ρb Ω, x k C n, k = 1,..., l, belong to V δ+ we conclude from 3.10 and 3.22 that dom V δ+ and ran V δ+ are dense in E T δ + G and E B δ + K, respectively. From the fact that E T δ + G, [, ] G is a Hilbert space we conclude that ran V δ+ and E B δ + K are nonnegative subspaces of the Pontryagin space E B δ + K, [, ]. Let us show that L 0 := { x E B δ + K [x, x] = 0 } is trivial. As E B δ + K is nonnegative we have L 0 [ ]E B δ + K and L 0 [ ]I E B δ + K, and therefore L 0 K [ ]. In view of 3.11 A λ 1 L 0 = B λ 1 L 0 holds for all λ ρa ρb Ω. Hence for x 0 L 0 and δ + = d 1, d 2 we conclude that E A δ + x 0 = lim η 0 lim ɛ 0 1 2πi d2 η d 1 +η A λ+iɛ 1 A λ iɛ 1 x 0 dλ 17

18 and E B δ + x 0 coincide. As E A δ + L 0 = E B δ + L 0 and E A δ + K, [, ] is a Hilbert space for x E B δ + L 0, x 0, we find [x, x] 0. Then by E B δ + L 0 = L 0 we conclude L 0 = {0}. The fact L 0 = {0} implies that the multivalued part mul V δ+ = { x E B δ + K 0 x V δ+ } of V δ+ is trivial. Hence V δ+ is a densely defined closed isometric operator. We claim that ran V δ+ is closed. As in the proof of [17, Theorem 6.2] one verifies that V δ+ is a bounded operator and therefore dom V δ+ = E T δ + G holds. From L 0 = {0} we also obtain that V δ+ is injective and another application of [17, Theorem 6.2] shows that the closed isometric operator V 1 δ + is bounded. Thus dom V 1 δ + = ran V δ+ is closed. As ran V δ+ = E B δ + K is a closed positive subspace of the Pontryagin space E B δ + K, [, ] we infer that E B δ + K is uniformly positive, i.e. E B δ + K equipped with the inner product [, ] is a Hilbert space. iv Let H be the orthogonal complement of E B δ + K in the Pontryagin space E B δ + K, [, ], 3.23 E B δ + K = E B δ + K [ +]H. H is a Pontryagin space with finite rank of negativity. From H[ ]E B δ + K and H[ ]I E B δ + K we obtain H K [ ]. By 3.11 the resolvents of A and B restricted to H coincide and by writing the projections E A δ + and E B δ + as strong limits of the resolvent of A and B, respectively, we see that E A δ + H and E B δ + H coincide. As above we obtain that H = E B δ + H is a Hilbert space and from 3.23 we conclude that E B δ + K, [, ] is a Hilbert space. As for any open interval δ + in +, δ + +, such that E B δ + is defined the spectral subspace E B δ + K, [, ] is a Hilbert space it follows that the open interval + is of positive type with respect to B. In fact, let ξ + σb and choose an open interval δ + with ξ δ + such that δ + + and the boundary points of δ + are of positive type with respect to B. If x n yn B is a sequence with x n = 1 and y n ξx n 0 for n then B I EB δ + K 2 1 ξ L I EB δ + K and lim I E Bδ + y n ξx n = 0 n 18

19 imply I E B δ + x n 0 and E B δ + x n 1 for n. As E B δ + K equipped with the inner product [, ] is a Hilbert space we have lim inf n [x n, x n ] = lim inf n [E Bδ + x n, E B δ + x n ] > 0, that is, ξ is of positive type with respect to B. Hence + is of positive type with respect to B. Analogously one verifies that an open interval with endpoint µ Ω R which is of negative type with respect to A contains an open interval with endpoint µ which is of negative type with respect to B. Therefore we have shown that for every point µ Ω R there exists an open connected neighbourhood I µ in R such that both components of I µ \{µ} are of the same sign type with respect to A and B. It remains to verify that B belongs to S Ω R. As α is a definitizable function in Ω and A is definitizable over Ω no point of Ω R is an accumulation point of nonreal poles of α and nonreal spectrum of A in Ω\R. Hence by 3.12 the nonreal spectrum of B in Ω\R does not accumulate to points in Ω R and by 3.6 the set σb Ω\R consists of isolated points which are poles of the resolvent of B. Now the growth properties of α see Definition 2.6 and the resolvent of A imply B S Ω R. Therefore B is definitizable over Ω and Theorem 3.2 is proved. 4. Self-adjoint extensions of symmetric operators and direct sums of Sturm-Liouville operators In this section we apply the general perturbation results from Section 3 to self-adjoint extensions of symmetric operators and relations of finite defect. As an example we consider direct sums of symmetric Sturm-Liouville operators with the indefinite weight sgn x where the self-adjoint extensions are not definitizable but turn out to be locally definitizable over C. First we recall some necessary definitions and the notion of boundary value spaces and associated Weyl functions Self-adjoint extensions of symmetric operators and relations of finite defect. Let K be a separable Krein space, let J be a corresponding fundamental symmetry and let S CK be a closed symmetric relation in K. We say that S is of defect m N { }, if both deficiency indices n ± JS = dim ker JS λ, λ C ±, of the symmetric relation JS in the Hilbert space K, [J, ] are equal to m. With the help of the von Neumann formulas for a closed symmetric relation 19

20 in a Hilbert space see e.g. [13, 2.3] one can verify without difficulty that this is equivalent to the fact that there exists a self-adjoint extension of S in K and that each self-adjoint extension à of S in K satisfies dimã/s = m. For the description of the self-adjoint extensions of closed symmetric relations we use the so-called boundary value spaces. Definition 4.1. Let S be a closed symmetric relation in the Krein space K. We say that {G, Γ 0, Γ 1 } is a boundary value space for S + if G,, is a Hilbert space and there exist linear mappings Γ 0, Γ 1 : S + G such that Γ := Γ 0 Γ 1 : S + G G is surjective, and the relation 4.24 holds for all ˆf = f f, ĝ = g g S +. [f, g] [f, g ] = Γ 1 ˆf, Γ0 ĝ Γ 0 ˆf, Γ1 ĝ If S is closed symmetric relation in K and à CK is a self-adjoint extension of S with ρã, then there exists a boundary value space {G, Γ 0, Γ 1 } for S + such that à coincides with ker Γ 0 see [11]. For basic facts on boundary value spaces and further references see e.g. [10, 11, 12, 13]. We recall only a few important consequences. Let S be a closed symmetric relation and assume that there exists a boundary value space {G, Γ 0, Γ 1 } for S +. Then 4.25 A 0 := ker Γ 0 and A 1 := ker Γ 1 are self-adjoint extensions of A. The mapping Γ = Γ 0 Γ 1 induces, via 4.26 A Θ := Γ 1 Θ = { ˆf S + Γ ˆf Θ }, Θ CG, a bijective correspondence Θ A Θ between CG and the set of closed extensions A Θ S + of S. In particular 4.26 gives a one-to-one correspondence between the closed symmetric self-adjoint extensions of S and the closed symmetric resp. self-adjoint relations in G. If Θ is a closed operator in G, then the corresponding extension A Θ of S is determined by 4.27 A Θ = ker Γ 1 ΘΓ 0. Let N λ := kers + λ = ran S λ [ ], λ rs, be the defect subspace of S and set ˆN λ := { } f f λf Nλ. Now we assume that the self-adjoint relation A 0 in 4.25 has a nonempty resolvent set. Then for λ ρa 0 the adjoint S + is the direct sum of A 0 and 20

21 ˆN λ. The functions and 4.28 λ γλ := Γ 0 ˆN λ 1 LG, K, λ ρa 0, λ Mλ := Γ 1 Γ0 ˆN λ 1 LG, λ ρa 0, are holomorphic on ρa 0 and are called the γ-field and Weyl function corresponding to S and {G, Γ 0, Γ 1 }. We remark that for a fixed λ 0 ρa 0 and all λ ρa 0 the Weyl function M can be written in the form 4.29 Mλ = Re Mλ 0 + γλ 0 + λ Re λ 0 + λ λ 0 λ λ 0 A 0 λ 1 γλ 0. With the help of the Weyl function the spectral properties of the closed extensions of S can be described. If Θ CG and A Θ is the corresponding extension of S via 4.26 then a point λ ρa 0 belongs to ρa Θ σ i A Θ, i = p, c, r if and only if 0 belongs to ρθ Mλ resp. σ i Θ Mλ, i = p, c, r and the well-known formula A Θ λ 1 = A 0 λ 1 + γλ Θ Mλ 1 γλ + holds for all λ ρa 0 ρa Θ see e.g. [11]. In the special case that S is of defect one the self-adjoint extensions A Θ of S, A Θ ker Γ 0, in K can be parametrized with the real numbers Θ R. Therefore, in this case, all self-adjoint extensions of S have a nonempty resolvent set if the scalar Weyl function is not equal to a constant. The following theorem is an immediate consequence of Theorem 3.1, Theorem 3.2 and the fact that the difference of the resolvents of two self-adjoint extensions of a symmetric relation of finite defect is a finite rank operator. Theorem 4.2. Let S be a closed symmetric relation in the Krein space K and assume that the defect of S is finite. Then the following holds. i If there exists a self-adjoint extension A of S in K which is definitizable over Ω, then every self-adjoint extension à of S in K with ρã Ω is definitizable over Ω. ii If A is a self-adjoint extension of S in K which is definitizable over Ω and Ω R is an open interval with endpoint µ Ω R and is of positive type negative type with respect to A, then for every self-adjoint extension à of A with ρã Ω there exists an open interval,, with endpoint µ such that is of positive type resp. negative type with respect to Ã. 21

22 iii If there exists a self-adjoint extension A of S in K which is of type π + type π over Ω, then every self-adjoint extension à of S in K with ρã Ω is of type π + type π over Ω Direct sums of second order differential operators. In this section we investigate the spectral properties of direct sums of regular and singular Sturm-Liouville operators with the indefinite weight sgn x. The following notation will be useful. If K, [, ] K and H, [, ] H are Krein spaces the elements of K H will be written in the form {k, h}, k K, h H. The direct sum of a linear operator S in K and a linear operator T in H will be denoted by S T. If S and T are symmetric in K and H, respectively, then S T is symmetric in the Krein space K H, [, ], where [{k, h}, { k, h}] := [k, k] K + [h, h] H, k, k K, h, h H. Let in the following K, [, ] K be the Krein space L 2 R equipped with the inner product and let [f, g] K := fxgx sgn x dx, Sfx := sgn x f x f, g L 2 R, dom S := { f W 2,2 R f0 = f 0 = 0 }. Then S is a densely defined closed symmetric operator in K of defect two and the adjoint operator S + is given by S + fx = sgn x f x, dom S + = W 2,2 R W 2,2 R +, where R + := 0, and R :=, 0. A straightforward calculation shows that {C 2, Γ 0, Γ 1 }, where f0+ f0 f 4.30 Γ 0 ˆf := f 0+ f, ˆf = 0 S +, f and 4.31 Γ 1 ˆf := 1 2 f 0+ + f 0 f, ˆf = f0+ f0 S +, f is a boundary value space for S + and the self-adjoint extension A 0 = ker Γ 0 is the usual second order differential operator with the indefinite weight function x sgn x on R. 22

23 Besides the Krein space K we consider the Krein space H := L 2 a, b, [, ] H, where < a < 0 < b < and [, ] is defined by [h, k] H := b a hxkx sgn x dx, h, k L 2 a, b. Let p 1, q L 1 a, b be real functions and assume that p > 0 is fulfilled. We consider the densely defined closed symmetric operator T, T hx := sgn x pxh x + qxhx, { } dom T := h L 2 h, ph a, b W 1,2 a, b, ph + qh L 2 a, b ha = hb = h0 = ph, 0 = 0 in H. Then T has defect two and the operator T + is given by T + hx = sgn x pxh x + qxhx, { } dom T + = h L 2 h, ph a, b W 1,2 a, 0 W 1,2 0, b, ph + qh L 2. a, b, ha = hb = 0 Here we choose {C 2, Γ 0, Γ 1 }, where h0 ph Γ 0ĥ := and Γ h0+ 1ĥ := ph 0+, ĥ = h T +, h as boundary value space for T +. We remark that the self-adjoint extension B Φ of T in H, [, ] H corresponding to the self-adjoint relation { x, x } Φ = x, y C y, y CC 2 via 4.26 is the usual second order differential operator sgn x d p d + q dx dx in L 2 a, b with Dirichlet boundary conditions. Theorem 4.3. Let S and T be the symmetric differential operators in the Krein spaces K and H from above and let {C 2, Γ 0, Γ 1 }, A 0 = ker Γ 0, and {C 2, Γ 0, Γ 1 }, B 0 = ker Γ 0, be the boundary value spaces from and 4.32 and denote the corresponding Weyl functions by M and τ, respectively. Then the following assertions i-iii hold. i All canonical self-adjoint extensions of S and T in the Krein spaces K and H, respectively, are definitizable over C. 23

24 ii The self-adjoint operator A 0 B 0 in the Krein space K H is definitizable over C\{ } and σa 0 B 0 coincides with R. The interval 0, is of type π + and the interval, 0 is of type π with respect to A 0 B 0. iii If Θ CC 4 is a self-adjoint relation such that 0 ρmλ τλ Θ for some λ C\R, then the self-adjoint differential operator ÃΘ e = S+ T + dom ÃeΘ, dom {{f, ÃeΘ = Γ0 ˆf, Γ h} K H 0 ĥ Γ 1 ˆf, Γ 1 ĥ Θ, ˆf = f } S + f, ĥ = h T + h in K H is definitizable over C\{ } and the interval 0,, 0 is of type π + resp. type π with respect to Ãe Θ. Proof. i As S is a densely defined symmetric operator in K, [, ] K and A 0 is a nonnegative self-adjoint operator with σa 0 = R see e.g. [9] it follows from [8, Proposition 1.1] that all self-adjoint extensions A Θ, Θ CC 2, of S in K are definitizable over C. Denote by p 1 p 2 and q 1 q 2 the restrictions of the functions p and q onto the interval a, 0 resp. 0, b. Then the self-adjoint extension B 0 = ker Γ 0 of T in H is a fundamentally reducible operator as it coincides with the direct sum of the self-adjoint realizations of the regular Sturm-Liouville differential expressions d p dx 1 d q dx 1 and d p dx 2 d + q dx 2, in L 2 a, 0,, and L 2 0, b,, corresponding to Dirichlet boundary conditions. Hence σb 0 is real and consists only of eigenvalues with one or two-dimensional eigenspaces accumulating only to and. Here the assumptions p 1, p 2 > 0 imply that there are only finitely many eigenvalues belonging to σ ++ B 0 0, and σ B 0, 0 cf. [26, 36]. Therefore the hermitian form [B 0, ] defined on dom B 0 has finitely many positive squares and it follows again from [8, Proposition 1.1] that all self-adjoint extensions B Φ, Φ CC 2, are definitizable. ii Since A 0 and B 0 are definitizable they belong to the class S R and therefore A 0 B 0 is also in the class S R. From σa 0 σb 0 = R we obtain σa 0 B 0 = R. In order to see that A 0 B 0 is definitizable over C\{ } we have to check that for every point µ R there is an open interval I µ R, µ I µ, such that both components of I µ \{µ} are of definite type with respect to A 0 B 0. This follows from the nonnegativity of A 0, hence 0,, 0 is of positive type resp. negative type with respect to A 0, and the fact that σb 0 consists of eigenvalues accumulating only to and. 24

25 Let δ be an open interval such that δ 0, and the boundary points of δ in R are no eigenvalues of B 0. As the spectral subspace E A0 δk, [, ] K is a Hilbert space and E B0 δh, [, ] H is a finite dimensional Pontryagin space we conclude that 0, is of type π + with respect to A 0 B 0. A similar argument shows that, 0 is of type π with respect to A 0 B 0. iii It is easy to see that Γ 0 { ˆf, ĥ} := Γ0 ˆf and Γ 0ĥ Γ 1 { ˆf, ĥ} := Γ1 ˆf, Γ 1ĥ { ˆf, ĥ} := { f S + f, h } T + h S + T +, defines a boundary value space {C 4, Γ 0, Γ 1 } for S + T + with ker Γ 0 = A 0 B 0 and corresponding Weyl function Mλ λ = Mλ τλ LC 0 τλ 4, λ ρa 0 B 0. Now assertion iii follows from Theorem 4.2. Let S and T be the symmetric differential operators from above and let {C 2, Γ 0, Γ 1 } and {C 2, Γ 0, Γ 1} be the boundary value spaces from and By 4.29 and Theorem 2.7 the Weyl functions M and τ corresponding to {C 2, Γ 0, Γ 1 } and {C 2, Γ 0, Γ 1 }, respectively, are definitizable functions in C see Section 2.3 and [22] and the function 4.33 is definitizable in C. Here M can be calculated explicitely and also the structure of τ is known. Indeed, if + denotes the branch of defined in C with a cut along [0,, 0] and fixed by Im λ > 0 for λ [0, and λ 0 for λ [0, resp. Re λ > 0 for λ, 0] and Im λ 0 for λ, 0], then for λ C\R the defect subspace N λ = kers + λ is spanned by the functions f λ x := { exp i + λx x > 0 0 x < 0 and g λ x := { 0 x > 0 exp λx x < 0. Hence with ˆf λ = f λ λf λ and ĝλ 1 i + λ 2 1, Γ0 ĝ λ = 1 λ and Γ1 ĝ λ = 1 λ 2 1 M corresponding to the boundary value space {C 2, Γ 0, Γ 1 } is given by Mλ = 1 i + λ λ = g λ λg λ we have Γ0 ˆfλ = 1 i + λ, Γ1 ˆfλ = and therefore the Weyl function i + λ 1 λ i + λ + λ i λ + λ 1 2, λ C\R. Similarly kert + λ, λ ρb 0, is spanned by some functions h λ k λ, which vanish on the interval 0, b resp. a, 0. It is not difficult to see that 25

26 there exist scalar Nevanlinna functions N 1 and N 2, hn 1 hn 2 = ρb 0, such that the Weyl function τ corresponding to {C 2, Γ 0, Γ 1} has the form N1 λ 0 τλ =, λ ρb 0 N 2 λ An example for a self-adjoint extension ÃeΘ of a direct sum of differential operators with σ p Ãe Θ = C. In the following we will give a simple example of a direct sum S T of two differential operators S and T where a certain self-adjoint extension has an empty resolvent set. In the Hilbert space K := L 2 α, β,,, < α < β <, we consider the symmetric second order differential operator Sfx := f x, dom S := { f L 2 α, β f W 2,2 α, β, fα = f α = fβ = 0 }, the adjoint operator S, S fx = f x, dom S = { f L 2 α, β f W 2,2 α, β, fβ = 0 }, and we choose {C, Γ 0, Γ 1 }, Γ 0 ˆf := fα, Γ1 ˆf := f α, ˆf = f S f, as a boundary value space for S. Let kers λ = sp {f λ }, λ C\R. Then the Weyl function M corresponding to {C, Γ 0, Γ 1 } is given by Mλ = Γ 1 ˆf λ Γ 0 ˆfλ, ˆfλ = fλ λf λ, λ C\R. We equip L 2 α, β with the negative definite inner product [, ] defined by [g, h] := g, h, g, h L 2 α, β, and denote the corresponding Krein space by H. The differential operator T hx := h x, dom T = dom S, is symmetric in H and the adjoint operator is T + hx = h x, dom T + = dom S. Here {C, Γ 0, Γ 1 }, Γ 0ĥ := h α, Γ 1ĥ := hα, ĥ = h S + h, is a boundary value space for T + and the corresponding Weyl function τ is given by τλ = Γ 1 ˆf λ Γ 0 ˆf λ = Γ 0 ˆf λ Γ 1 ˆfλ = 1 Mλ, λ C\R. 26

27 As in the proof of Theorem 4.3 iii we define the boundary value space {C 2, Γ 0, Γ 1 } for S T + by Γ 0 { ˆf, ĥ} := Γ0 ˆf and Γ 0ĥ Γ 1 { ˆf, ĥ} := Γ1 ˆf, Γ 1ĥ { ˆf, ĥ} := { f S f, h } T + h. Note that the selfadjoint operator ker Γ0 is definitizable over C\{ }. Now the corresponding Weyl function M is λ Mλ Mλ 0 = 1 0, λ C\R. Mλ The self-adjoint extension Ãe Θ, Θ := LC 2, of S T in the Krein space K H via is given by à eθ = S T + dom Ãe Θ dom ÃeΘ = { {f, h} dom S dom T + fα = hα, f α = h α } and we have σ p ÃeΘ = C since the function λ det Mλ Θ is identically equal to zero. Acknowledgements. The author thanks Peter Jonas for his fruitful comments and his valuable help in the preparation of this manuscript. References [1] T.Ya. Azizov, P. Jonas: On locally definitizable matrix functions, Preprint TU Berlin, Institute of Mathematics, [2] T.Ya. Azizov, P. Jonas, C. Trunk: Spectral points of type π + and type π of selfadjoint operators in Krein spaces, J. Funct. Anal , [3] J. Behrndt: A class of abstract boundary value problems with locally definitizable functions in the boundary condition, Operator Theory: Advances and Applications 163 Birkhäuser Verlag Basel 2005, [4] J. Behrndt, P. Jonas: On compact perturbations of locally definitizable selfadjoint relations in Krein spaces, Integral Equations Operator Theory , [5] J. Behrndt, P. Jonas: Boundary value problems with local generalized Nevanlinna functions in the boundary condition, to appear in Integral Equations Operator Theory. [6] J. Behrndt, C. Trunk: Sturm-Liouville operators with indefinite weight functions and eigenvalue depending boundary conditions, to appear in J. Differential Equations. [7] P. Binding, R. Hryniv, H. Langer, B. Najman: Elliptic eigenvalue problems with eigenparameter dependent boundary conditions, J. Differential Equations ,

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